If Map projection is different from Drawing projection then there is displacement.
I've done this a zillion times and have never seen that issue when a map projection is different than the drawing, for example, the photo below, where the map is in pseudo mercator and the drawing is in lat/lon. Works perfectly every time. (the photo is a combination of two screenshots, one showing the Component pane and the other the "live action" editing..) To make photos in posts bigger, just click on them.
Could you post a screenshot of the Component panel of the Contents pane for your map with drawing layer?
If there is a problem with the way a coordinate system has been assigned, figuring out errors in workflow requires detailed, step by step descriptions of how you created what you are working with. Problems are frequently caused by the key differences in dialogs such as Assign Initial Coordinate System and Change Coordinate System.
Ah... one more idea. In cases of extreme mismatch between coordinate systems, like trying to show a highly local coordinate system overlaid on a "whole Earth" coordinate system like pseudo mercator, you can, indeed, get crazy effects. Normally such craziness happens if coordinate systems are wrongly assigned, but they can happen when coordinate systems are correctly assigned but a wildly inappropriate coordinate system has been used, or with wildly inappropriate parameters.
For example, Lambert Conformal Conic (LCC) is a sensible coordinate system but if you project a drawing into LCC centered on Kansas and you start clicking away at it as a layer in a map that uses Pseudo Mercator and is panned and zoomed to Zimbabwe, you're going to get some strange effects.
If you tell us more about what you've done, all details, we can help zero in beyond such very general discussion.